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As we already introduced in the previous section, an exciton is the bound electron- hole pair in semiconductors. Thus, at low densities, exciton are hydrogen-like bosonic particles with two important characteristic. First, the Bohr radius is large because excitons present a really small reduced mass (≈me/20) and, secondly, their binding

energy energy is small since the Coulomb interaction is significantly reduced by the dielectric constant presents in the semiconductor. The first of the first characteristic makes the excitons a suitable physical system for observing the condensation at high temperatures, i.e. order of 1 K. The comparison of excitons to hydrogens are shown in the Table 3.1. Excitons also form in an organic semiconductors where Coulumbic

Excitons Hydrogen reduced mass ∼0.058me ∼1.0me

Bohr radius ∼12nm ∼0.053nm

binding energy ∼5.0meV ∼13.6eV

Table 3.1: Comparison between GaAs exciton parameters in the reduced mass ap- proximation and those of the Hydrogen atom.

interactions are stronger than inorganic semiconductors due its the dielectric con- stant therefore excitons are tightly bound which result atomic scale object, Frenkel excitons.

3.2.1 Bulk semiconductor excitons

Schematics of an typical direct gap semiconductor band structure is shown in Figure. 3.1, where the locale minima of conducting and maximum of valence band are exactly at same momenta, consists of a filled conduction electron band separated from an empty valence electron band through a well defined gap Egap. The absorption of a

photon with energy close to the semiconductor gap can promoted an electron from the conduction to the valence band. The electron and hole appearing in the system , located at the valence and conduction band respectively, may created a bound state, the exciton, by minimizing their Coulomb attraction. As we already mentioned, in some semiconductor materials like GaAs, the binding energy of such a pair is small compare to the usual one between electrons, since the Coulomb interaction is screened by the valence electrons. These excitons are extended over many lattice spacings, which is one of the principal characteristic of the so-called Wannier-Mott

Figure 3.1: Schematics of an idealised bandstructure for a direct gap semiconductor which consist of a completely filled valence band (blue) and an empty conduction band (orange). The electron excitations in the direct gap semiconductors illustrated by the arrow, in the absence of electron-electron in teractions, where an electron is exciting from the valence band to the conduction band, leaving a positively charged hole in the valence band.

excitons, their dynamics can be described with the following effective Hamiltonian: ˆ HX= p2_h 2m∗_h + p2_e 2m∗ e − e 2 4π∗_|r h−re| . (3.1) We observe that the electron and hole present effective masses m∗_e and m∗_h respec- tively, which are considered to be isotropic and with values given by the specific form of the conduction and valence dispersion relations. We observe also the effective di- electric constant∗, which considers the screening of the interaction with by the rest of the electron at the conduction band. As usual, we can calculated the effective massmX of the exciton with the relation 1/mX = 1/m∗e+m∗h; which, as we alredy

mentioned, can be 20 times smaller than the electron mass for usual experimental setups. The eigenvalues of the Hamiltonian (3.1) can be obtained following the same techniques than in the hydrogem atom case. Then, the dispersion relation for the excitons reads as:

EX(q) =Egap− ERy n2_X + ~2q2 2mX , (3.2) whereEgapis the gap of the semiconductor,ERy is the excitonic Rydberg of binding

for excitons andnX = 1,2, ...is the principal quantum number. Therefore, we have

a physical illustration about excitons in semiconductors which is shown in the Figure (3.2)

Figure 3.2: Cartoon of a hydrogen like discretized energy bands of excitons in a direct gap semiconductor given by equation (3.2). The most upper dashed line is the energy level of E(∞), is the minimum energy of the continuum state -upper bound of discrete energy level and which is equal to band gap Eg.

3.2.2 Quantum well excitons

As we already mentioned in the introduction of the current chapter, the consideration of the excitons in confined structures overcomes some of the obstacles presented in considering excitons in bulk semiconductors.

Particularly, this is the case of the two-dimensional quantum wells (QW). A QW is thin layered semiconductor sandwiched between other two semiconductor ”barrier” layers of wider bandgap material (see Figure 3.3). In such plataforms, the energy levels of the excitons are quantized. Current advances in material growth techniques, specifically of molecular beam epitaxy, allow to obtain QW with thick- ness comparable to the exciton Bohr radius. In such cases, the dynamics of the excitons is confined to the QW plane. At first considered by Lozovik, Yudson and Shevchenko [64–66] the lifetime of the exciton can be significantly increased by con- fining electron and holes in separated two-dimensional layers, i.e. by considering the implementation of coupled quantum wells (see Figure 3.3). Indirect excitons, namely the ones formed in coupled quantum wells, exhibit more favourable properties to- wards condensation, as the strong dipole interaction between them, which prevents from the formation of excitations and stabilizes the system. Thus, a substantial effort has been done towards the study, characteritzation and control of indirect excitons [71].

Figure 3.3: Scheme of excitons in coupled quantum wells. Coulombic electron and hole bound in the same wells are called direct excitons while if excitons are formed from electrons and holes located in different wells known as indirect excitons.

3.2.3 Quamtum well excitons coupling with light

The interaction between the QW excitons and the cavity photons conserves the total energy and the total in-plane momentum. Moreover, this coupling also conserves the total angular-momentum. Electron can have spin projection±1/2 along the growth axisz, whereas the holes in the valence band present spin projection±3/2 in such a direction. Therefore, the projection of the spin of the excitons in thezdirection can take the values±1,±2. The±2 excitons are optically inactive, they do not couple to light since in this process angular momentum won’t conserved, they are called dark excitons. Whereas the±1 excitons, the bright excitons, can couple to±1 polarized light, leading to polaritons.